Let $\Omega$ be the set of all $2 \times 2$ matrices of the form $\alpha=a I+b J+c K+d L$, where $a, b, c, d$ are in $\mathbf{R}$, and

$$I=\left(\begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array}\right), J=\left(\begin{array}{cc} i & 0 \\ 0 & -i \end{array}\right), K=\left(\begin{array}{cc} 0 & 1 \\ -1 & 0 \end{array}\right), L=\left(\begin{array}{cc} 0 & i \\ i & 0 \end{array}\right) \quad\left(i^{2}=-1\right) .$$

Prove that $\Omega$ is closed under multiplication and determine its dimension as a vector space over $\mathbf{R}$. Prove that

$$(a I+b J+c K+d L)(a I-b J-c K-d L)=\left(a^{2}+b^{2}+c^{2}+d^{2}\right) I$$

and deduce that each non-zero element of $\Omega$ is invertible.